Geodesic
2-Local Isometries on Spaces of Lipschitz Functions
Canadian mathematical bulletin, Tome 54 (2011) no. 4, pp. 680-692

Voir la notice de l'article provenant de la source Cambridge

DOI

Let $(X,\,d)$ be a metric space, and let $\text{Lip(}X\text{)}$ denote the Banach space of all scalar-valued bounded Lipschitz functions $f$ on $X$ endowed with one of the natural norms $$\left\| f \right\|\,=\,\max \{{{\left\| f \right\|}_{\infty }},\,L(f)\}\,\,\text{or}\,\,\left\| f \right\|\,=\,{{\left\| f \right\|}_{\infty }}\,+\,L(f),$$ where $L(f)$ is the Lipschitz constant of $f$ . It is said that the isometry group of $\text{Lip(}X\text{)}$ is canonical if every surjective linear isometry of $\text{Lip(}X\text{)}$ is induced by a surjective isometry of $X$ . In this paper we prove that if $X$ is bounded separable and the isometry group of $\text{Lip(}X\text{)}$ is canonical, then every 2-local isometry of $\text{Lip(}X\text{)}$ is a surjective linear isometry. Furthermore, we give a complete description of all 2-local isometries of $\text{Lip(}X\text{)}$ when $X$ is bounded.
DOI : 10.4153/CMB-2011-025-5
Mots-clés : 46B04, 46J10, 46E15, isometry, local isometry, Lipschitz function 
Jiménez-Vargas, A.; Villegas-Vallecillos, Moisés. 2-Local Isometries on Spaces of Lipschitz Functions. Canadian mathematical bulletin, Tome 54 (2011) no. 4, pp. 680-692. doi: 10.4153/CMB-2011-025-5
@article{10_4153_CMB_2011_025_5,
     author = {Jim\'enez-Vargas, A. and Villegas-Vallecillos, Mois\'es},
     title = {2-Local {Isometries} on {Spaces} of {Lipschitz} {Functions}},
     journal = {Canadian mathematical bulletin},
     pages = {680--692},
     year = {2011},
     volume = {54},
     number = {4},
     doi = {10.4153/CMB-2011-025-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-025-5/}
}
TY  - JOUR
AU  - Jiménez-Vargas, A.
AU  - Villegas-Vallecillos, Moisés
TI  - 2-Local Isometries on Spaces of Lipschitz Functions
JO  - Canadian mathematical bulletin
PY  - 2011
SP  - 680
EP  - 692
VL  - 54
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-025-5/
DO  - 10.4153/CMB-2011-025-5
ID  - 10_4153_CMB_2011_025_5
ER  - 
%0 Journal Article
%A Jiménez-Vargas, A.
%A Villegas-Vallecillos, Moisés
%T 2-Local Isometries on Spaces of Lipschitz Functions
%J Canadian mathematical bulletin
%D 2011
%P 680-692
%V 54
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-025-5/
%R 10.4153/CMB-2011-025-5
%F 10_4153_CMB_2011_025_5

Cité par Sources :